8. \( P(7,2) \)

To calculate \( P(7,2) \), also known as a permutation, you use the formula: \[ P(n, r) = \frac{n!}{(n-r)!} \] Plugging in the values, we have: \[ P(7,2) = \frac{7!}{(7-2)!} = \frac{7!}{5!} = \frac{7 \times 6 \times 5!}{5!} = 7 \times 6 = 42 \] So, \( P(7,2) = 42 \). In the realm of permutations, understanding how many ways to arrange a subset from a larger set can help in various scenarios, from choosing teams to scheduling tasks! Remember, order matters here, so every arrangement is unique—because choosing "A and B" isn't the same as "B and A". To avoid common mistakes, ensure you distinguish between permutations and combinations. While permutations count the order, combinations simply focus on the selection. When you’re unsure, ask yourself if the order affects the outcome!